If it’s an entrance exam for a university in America (and it clearly is), only the first option (with “be”) is correct. This is the present subjunctive (you can Google it for more information), and its proper use is still required in educated American English. In casual conversation, people (no one that I know, however) may simply use the indicative, although in the sentence you’ve given, I can’t imagine any American not using the subjunctive. British speakers, however, often don’t bother to use the subjunctive, and you might come across the version with “are” (although “be” is still considered to be the “correct” version). I doubt, however, that a British exam would test for your knowledge of the subjunctive these days.

Grammatically is go with the first one, "be" because after the adjectives that show necessity, we use the simple form is the verb. But English speakers also use the simple present form too. But as I said, if it's in a test, then the correct grammar is using the subjective.

It's the second one because in the first sentence you would have to add a "to" before the "be" and since there is another to after, it makes the sentence redundant.

Both are correct and both are a subjunctive form of the verb "to be." But also in both cases it would be better to use the simple present tense of the verb, "to limit." "It's necessary to limit classes to twenty students." Here you are more direct and and seem less bureaucratic.

The first one seems more accurate.